LMFDB - Newform orbit 1216.2.n.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,2,Mod(255,1216)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1216.255");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1216 = 2^{6} \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1216.n (of order $6$ , degree $2$ , not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$9.70980888579$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x + 1$ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 304)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$ -expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$ -expansion are expressed in terms of a primitive root of unity $\zeta_{6}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + ( - \zeta_{6} + 1) q^{3} + ( - 3 \zeta_{6} + 3) q^{5} + ( - 4 \zeta_{6} + 2) q^{7} + 2 \zeta_{6} q^{9} + ( - 4 \zeta_{6} + 2) q^{11} + ( - 3 \zeta_{6} + 6) q^{13} - 3 \zeta_{6} q^{15} + (3 \zeta_{6} - 3) q^{17} + \cdots + ( - 4 \zeta_{6} + 8) q^{99} +O(q^{100})$ q + (-z + 1) * q^3 + (-3z + 3) q^5 + (-4z + 2) q^7 + 2z q^9 + (-4z + 2) q^11 + (-3z + 6) q^13 - 3z q^15 + (3z - 3) q^17 + (2z + 3) q^19 + (-2z - 2) q^21 + (3z - 6) q^23 - 4z q^25 + 5 * q^27 + (5z - 10) q^29 + 4 * q^31 + (-2z - 2) q^33 + (-6z - 6) q^35 + (-6z + 3) q^39 + (5z + 5) q^41 + (-7z - 7) q^43 + 6 * q^45 + (-z + 2) * q^47 - 5 * q^49 + 3z q^51 + (z - 2) * q^53 + (-6z - 6) q^55 + (-3z + 5) q^57 + (3z - 3) q^59 + 7z q^61 + (-4z + 8) q^63 + (-18z + 9) q^65 - 5z q^67 + (6z - 3) q^69 + (9z - 9) q^71 + (7z - 7) q^73 - 4 * q^75 - 12 * q^77 + (-7z + 7) q^79 + (z - 1) * q^81 + (4z - 2) q^83 + 9z q^85 + (10z - 5) q^87 + (-5z + 10) q^89 - 18z q^91 + (-4z + 4) q^93 + (-9z + 15) q^95 + (5z + 5) q^97 + (-4z + 8) q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q + q^{3} + 3 q^{5} + 2 q^{9} + 9 q^{13} - 3 q^{15} - 3 q^{17} + 8 q^{19} - 6 q^{21} - 9 q^{23} - 4 q^{25} + 10 q^{27} - 15 q^{29} + 8 q^{31} - 6 q^{33} - 18 q^{35} + 15 q^{41} - 21 q^{43} + 12 q^{45}+ \cdots + 12 q^{99}+O(q^{100})$ 2 * q + q^3 + 3 * q^5 + 2 * q^9 + 9 * q^13 - 3 * q^15 - 3 * q^17 + 8 * q^19 - 6 * q^21 - 9 * q^23 - 4 * q^25 + 10 * q^27 - 15 * q^29 + 8 * q^31 - 6 * q^33 - 18 * q^35 + 15 * q^41 - 21 * q^43 + 12 * q^45 + 3 * q^47 - 10 * q^49 + 3 * q^51 - 3 * q^53 - 18 * q^55 + 7 * q^57 - 3 * q^59 + 7 * q^61 + 12 * q^63 - 5 * q^67 - 9 * q^71 - 7 * q^73 - 8 * q^75 - 24 * q^77 + 7 * q^79 - q^81 + 9 * q^85 + 15 * q^89 - 18 * q^91 + 4 * q^93 + 21 * q^95 + 15 * q^97 + 12 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$ .

$n$	$191$	$705$	$837$
$\chi(n)$	$-1$	$\zeta_{6}$	$1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

\iota_m(\nu)

a_{2}

a_{3}

a_{4}

a_{5}

a_{6}

a_{7}

a_{8}

a_{9}

a_{10}

255.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0.500000

−

0.866025i

1.50000

−

2.59808i

−

3.46410i

1.00000

1.73205i

639.1

0.500000

0.866025i

1.50000

2.59808i

3.46410i

1.00000

−

1.73205i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
76.f	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1216.2.n.b		2
4.b	odd	2	1		1216.2.n.a		2
8.b	even	2	1		304.2.n.a	✓	2
8.d	odd	2	1		304.2.n.b	yes	2
19.d	odd	6	1		1216.2.n.a		2
24.f	even	2	1		2736.2.bm.h		2
24.h	odd	2	1		2736.2.bm.g		2
76.f	even	6	1	inner	1216.2.n.b		2
152.l	odd	6	1		304.2.n.b	yes	2
152.o	even	6	1		304.2.n.a	✓	2
456.s	odd	6	1		2736.2.bm.g		2
456.v	even	6	1		2736.2.bm.h		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
304.2.n.a	✓	2	8.b	even	2	1
304.2.n.a	✓	2	152.o	even	6	1
304.2.n.b	yes	2	8.d	odd	2	1
304.2.n.b	yes	2	152.l	odd	6	1
1216.2.n.a		2	4.b	odd	2	1
1216.2.n.a		2	19.d	odd	6	1
1216.2.n.b		2	1.a	even	1	1	trivial
1216.2.n.b		2	76.f	even	6	1	inner
2736.2.bm.g		2	24.h	odd	2	1
2736.2.bm.g		2	456.s	odd	6	1
2736.2.bm.h		2	24.f	even	2	1
2736.2.bm.h		2	456.v	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{2} - T_{3} + 1$ T3^2 - T3 + 1 acting on $S_{2}^{\mathrm{new}}(1216, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} - T + 1$ T^2 - T + 1
$5$	$T^{2} - 3T + 9$ T^2 - 3*T + 9
$7$	$T^{2} + 12$ T^2 + 12
$11$	$T^{2} + 12$ T^2 + 12
$13$	$T^{2} - 9T + 27$ T^2 - 9*T + 27
$17$	$T^{2} + 3T + 9$ T^2 + 3*T + 9
$19$	$T^{2} - 8T + 19$ T^2 - 8*T + 19
$23$	$T^{2} + 9T + 27$ T^2 + 9*T + 27
$29$	$T^{2} + 15T + 75$ T^2 + 15*T + 75
$31$	$(T - 4)^{2}$ (T - 4)^2
$37$	$T^{2}$ T^2
$41$	$T^{2} - 15T + 75$ T^2 - 15*T + 75
$43$	$T^{2} + 21T + 147$ T^2 + 21*T + 147
$47$	$T^{2} - 3T + 3$ T^2 - 3*T + 3
$53$	$T^{2} + 3T + 3$ T^2 + 3*T + 3
$59$	$T^{2} + 3T + 9$ T^2 + 3*T + 9
$61$	$T^{2} - 7T + 49$ T^2 - 7*T + 49
$67$	$T^{2} + 5T + 25$ T^2 + 5*T + 25
$71$	$T^{2} + 9T + 81$ T^2 + 9*T + 81
$73$	$T^{2} + 7T + 49$ T^2 + 7*T + 49
$79$	$T^{2} - 7T + 49$ T^2 - 7*T + 49
$83$	$T^{2} + 12$ T^2 + 12
$89$	$T^{2} - 15T + 75$ T^2 - 15*T + 75
$97$	$T^{2} - 15T + 75$ T^2 - 15*T + 75
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